Aperiodic tilings with one prototile and low complexity atlas matching rules

TL;DR

This paper introduces a constructive method to reduce the number of prototiles in tiling problems by replacing matching rules with a small atlas of patches, demonstrating its effectiveness with specific examples.

Contribution

It presents a novel approach to simplify tiling rules by using an atlas of patches, enabling the construction of a single aperiodic prototile and a pair of tiles for specific dimensions.

Findings

Single prototile can tile R^3 aperiodically

Pair of square tiles can tile R^2 aperiodically

Method reduces complexity of tiling rules

Abstract

We give a constructive method that can decrease the number of prototiles needed to tile a space. We achieve this by exchanging edge to edge matching rules for a small atlas of permitted patches. This method is illustrated with Wang tiles, and we apply our method to present via these rules a single prototile that can only tile $R^3$ aperiodically, and a pair of square tiles that can only tile $R^2$ aperiodically.